# Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system,

$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$

$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$

and define the function,

$$F(s_m) = x_1+x_2+\dots+x_n$$

For $n\geq3$, using an elliptic curve, it can be shown there are an infinite number of primitive solutions. For $n=4$, we have positive and primitive,

\begin{aligned} s_1 = 10,\, 13,\, 14,\, 44;\quad &F(s_1) = 3^4\\ s_2 = 54,\, 109,\, 202,\, 260; \quad &F(s_2) = 5^4\\ s_3 = 51,\, 65,\, 117,\, 159; \quad &F(s_3) = 2^3 7^2\\ s_4 = 99,\, 315,\, 797,\, 837; \quad &F(s_4) = 2^{11}\\ s_5 = 285,\, 371,\, 547,\, 845; \quad &F(s_5) = 2^{11}\\ s_6 = 815,\, 1297,\, 1781,\, 1939; \quad &F(s_6) = 2^3 3^6\\ s_7 = 259,\, 1307,\, 3485,\, 9349;\quad &F(s_7) = 2^6 3^2 5^2\\ \end{aligned}

with $s_6$ and $s_7$ found by MSE user oleg567.

Questions (for $n=4$):

1. Why do these (apparently smallest) solutions have $F(s_m)$ as a smooth number?
2. Is this really the complete list with all $x_i<10^4$? (These are the smallest if the list in this post is appropriately complete.)

Comment: For $n=3,5$, small solutions do not necessarily have $F(s_m)$ that are smooth numbers.

Edit: Using $s_1$, we have,

$$P_k(x)=(\color{blue}{10}x^2 - x + 88)^k + (\color{blue}{13}x^2 + 68x + 28)^k + (\color{blue}{14}x^2 - 68x + 26)^k + (\color{blue}{44}x^2 + x + 20)^k$$

For $k=3$, it is already a cube,

$$P_3(x) = 45^3(x^2+2)^3$$

For $k=2$, it a quartic polynomial to be made a square with a rational point, hence an elliptic curve,

$$P_2(x) = y^2$$

with initial points $x = 68,\,165/94$, etc. All of the $s_m$ can be used in a similar quadratic form identity. Thus, after removing denominators, the system has an infinite number of integer solutions.

However, we are interested in the behavior of the small positive solutions and which seem to be knowable only by semi-brute force searching.

• [there are an infinite number of primitive solutions]-->what do you mean by "primitive" ? – Duchamp Gérard H. E. Apr 26 '15 at 4:15
• If I didn't make any mistakes, this system of equations defines a rationally connected 3-dimensional projective variety. There's a decent chance that we can parametrize the rational solutions. – zeb Apr 26 '15 at 4:20
• @DuchampGérardH.E.: We can define "primitive" as the $x_i$ do not have a common factor. – Tito Piezas III Apr 26 '15 at 8:30
• For $n=4$, maybe someone has a fast and efficient code to find all positive and primitive solutions with $x_i$ below a bound, say $x_i<10^4$. I am not yet 100% certain the list above is complete. – Tito Piezas III Apr 26 '15 at 8:44
• To me it also seems that the variety defined by Tito's equations is $3$-dimensional: it is a complete intersection of a quadric and a cubic hypersurface in $\mathbb{P}^5$, isn't it? – RP_ Apr 26 '15 at 10:36